I think people may be overselling this a bit. It is true that antimatter bears certain resemblances to matter traveling "backwards in time" when the math is examined, but the mental images conjured up by a plain-English reading of that phrase are almost entirely wrong. It's not like you tap the antimatter and the motion goes backward in time. I hate using metaphors, but it's somewhat more like you have a guy walking forward on the road, but if you hit him violently enough he'll end up turned around, walking back-first but in the same direction. Yeah, he's walking backwards through time! ime! ime! ime! but it's not like you can actually send messages backwards or anything; give him a message and he's still walking in the positive-time direction. There's a lot of cancellation of the negative term that goes on and he's still dancing to the tune of the same arrow-of-time as the rest of us.
I'm pretty pessimistic that antimatter will be repulsed by gravity, because you would suddenly have a term that would have to appear for the potential energy of the now-flipped gravity field. It makes much more sense for it to be affected normally. It's an interesting question that we might as well look at, on the off chance that the science will be wrong (which is when it advances, after all), but I wouldn't hold your breath.
I don't fully understand your analogy. How exactly is that situation "backwards in time" if it still is moving forwards? Where could I find more info on this?
Actually, it sounds like you got it perfectly. There is legitimately a way in which he's going "backwards", it's just not the way you would naturally think of if I just told you he's going "backwards". The backwards-through-time thing is one way of reading the math, but it doesn't mean what you think if you just read what the English says. Antimatter is way less interesting than "backwards in time" can make it sound. It's interesting, but not "kill your grandfather before your father is born" sort of interesting, just "fun particle physics" interesting.
I'm pretty pessimistic that antimatter will be repulsed by gravity, because you would suddenly have a term that would have to appear for the potential energy of the now-flipped gravity field. It makes much more sense for it to be affected normally. It's an interesting question that we might as well look at, on the off chance that the science will be wrong (which is when it advances, after all), but I wouldn't hold your breath.